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AMC 12 only

Law of Sines & Cosines

Solving triangles that aren't right triangles — and knowing which law fits which combination of known sides and angles.

Law of Sines

asin⁡A=bsin⁡B=csin⁡C\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}

Use this when you know an angle-side opposite pair, plus one more piece of information (AAS, ASA, or SSA).

Law of Cosines

c2=a2+b2−2abcos⁡Cc^2 = a^2+b^2-2ab\cos C

Use this when you know two sides and the included angle (SAS), or all three sides (SSS). Notice that when C=90°C=90°, this reduces exactly to the Pythagorean theorem — the Law of Cosines is a strict generalization of it.

The SSA ambiguous case

Given two sides and a non-included angle, the triangle isn’t always uniquely determined — there can be 0, 1, or 2 valid triangles, depending on whether the side opposite the given angle is long enough to “swing” into two different positions. Always check both possible angles when the Law of Sines gives you a sine value, and verify each candidate keeps the angle sum at 180°180°.

Area formulas

Area=12absin⁡C(SAS),Area=s(s−a)(s−b)(s−c)(SSS, Heron’s)\text{Area} = \dfrac12 ab\sin C \qquad\text{(SAS)}, \qquad \text{Area} = \sqrt{s(s-a)(s-b)(s-c)} \qquad\text{(SSS, Heron's)}

Worked example

Problem: A triangle has a=7a=7, b=9b=9, and included angle C=60°C=60°. Find cc and the triangle’s area.

Solution: By the Law of Cosines:

c2=72+92−2(7)(9)cos⁡60°=49+81−126(12)=130−63=67c^2 = 7^2+9^2-2(7)(9)\cos60° = 49+81-126\left(\tfrac12\right)=130-63=67

So c=67c=\sqrt{67}. Area, using the SAS formula:

Area=12(7)(9)sin⁡60°=632⋅32=6334\text{Area} = \dfrac12(7)(9)\sin60° = \dfrac{63}{2}\cdot\dfrac{\sqrt3}{2} = \dfrac{63\sqrt3}{4}

Practice problems

1.A triangle has A=40°A=40°, B=60°B=60°, and a=10a=10. Find bb (to a formula, not a decimal).

2.A triangle has sides a=6,b=8,c=10a=6, b=8, c=10. Find angle CC (opposite the longest side) using the Law of Cosines.

3.Find the area of a triangle with sides a=5,b=7a=5, b=7 and included angle 30°30°.

4.Explain why SSA is called the “ambiguous case” while SAS never is.

Trig Identities & EquationsLogarithms & Exponential Growth